Momentum space Feynman rules

Last updated Aug 5, 2023

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We can convert the Position space Feynman rules to momentum space Feynman rules by inserting the expressions for the propagators in momentum space.

The advantage is that the momentum space Feynman rules directly calculate the matrix element $\mathcal{M}$

Example: S-matrix in $\phi^3$

As first example we calculate the S-matrix for a massless scalar $\phi^3$ theory with $\mathcal{L} = -\frac{1}{2}\phi\Box\phi + \frac{g}{3!}\phi^3$.

We want to calculate the S-matrix for a process $x_1(p_i)\to x_2(p_f)$. To do so we use the LSZ reduction formula. $$ \begin{align*} \braket{f|S|i} = &\left[i\int d^4x_1 e^{-ip_1x_1}(\square_1)\right]\left[i\int d^4x_n e^{-ip_nx_n}(\square_n) \right] \\ &\times \braket{\Omega|T{\phi(x_1)\phi(x_n)}|\Omega} \end{align*} $$ As an example we only calculate a single contribution $\mathcal{T}1$ to the correlation function, e.g. the one loop diagram. The Position space Feynman rules give $$\mathcal{T}1 = -\frac{g^2}{2}\int d^4x \int d^4y D{1x}D^2{xy}D_{y2}$$ To simplify we insert the expression for the Feynman propagator in mometum space $$D_{xy} = \int \frac{d^4p}{(2\pi)^4}\frac{i}{p^2 + i\epsilon} e^{ip(x-y)}$$ Afterwards the integrals over the position of the interal vertices can be written as delta functions corresponding to the conservation of momentum at the vertices e.g. $\delta^4(-p_1 + p_3 +p_4)$. Using a delta function to execute one integral we get $$\mathcal{T}_1 = -\frac{g^2}{2}\int \frac{d^4k}{(2\pi)^4} \int \frac{d^4p_1}{(2\pi)^4}\int \frac{d^4p_2}{(2\pi)^4} e^{ip_1x_1} e^{-ip_2x_2} \frac{i}{p^2_1 + i\epsilon}\frac{i}{p^2_2 + i\epsilon}\frac{i}{(p_1 - k)^2 + i\epsilon}\frac{i}{k^2 + i\epsilon} (2\pi)^4 \delta^4(p_1-p_2)$$ Applying the LSZ reduction formula the derivatives $\Box_1$ and $\Box_2$ give factors of $p^2_i$ and $p^2_f$. $$\braket{f|S|i}=-\int d^4x_1 e^{-ip_ix_1}p^2_i \int d^4x_2 e^{ip_fx_2} p^2_f \mathcal{T}_1$$ After inserting $\mathcal{T}_1$ we get some $(2\pi)^4\delta$ functions from the $x$ integrals which cancel the $\frac{p^2_1}{p^2_1 + i\epsilon}$ factors to arrive at the contribution of the $S$ matrix element $$\braket{f|S|i} = -\frac{g^2}{2}\int \frac{d^4k}{(2\pi)^4}\frac{i}{(p_i - k)^2 + i\epsilon}\frac{i}{k^2 + i\epsilon} (2\pi)^4 \delta^4(p_i - p_f) + …$$ The propagators for external legs always cancel. Also the total momentum conserving $\delta$ will always appear and we always factor it out of the matrix element $\mathcal{M}$.

The calculations generalize to the momentum space feynman rules

Momentum space Feynman rules

1. Internal lines (not connected to external points) get a propagator $\frac{i}{p^2 - m^2 + i\epsilon}$
2. External lines don’t get propagators, they cancel in the LSZ reduction formula
3. Vertices get a factor $ig$ with $g$ the coupling of the interaction term in the Lagrangian
4. Momentum is conserved at each vertex
5. Integrate over undetermined 4-momenta (Internal momenta)
6. Sum over all possible diagrams

Note to properly define the direction of momenta so that at each vertex $\sum(p_\text{in} - p_\text{out})$.

Disconnected graphs (some external point not connected to some other external point) always factor out and normalize away.

Example: $\phi\phi\to\phi\phi$ in $\phi^3$

Take the lagrangian $$\mathcal{L} = -\frac{1}{2}\phi\Box\phi - \frac{1}{2}m^2\phi^2 + \frac{g}{3!}\phi^3$$ We want to calculate the cross section $$\frac{d\sigma}{d\Omega}(\phi\phi\to\phi\phi) = \frac{1}{64\pi^2E^2_\text{CM}}|\mathcal{M}|^2.$$ We denote $p_1,p_2$ the incoming momenta and $p_3,p_4$ the outgoing momenta. There are three possible diagrams at $\mathcal{O}(g^2)$ that contribute to $\mathcal{M}$:

1. s-channel: $s=(p_1+p_2)^2$ $$i\mathcal{M}_s = (ig)\frac{i}{(p_1 + p_2)^2 -m^2 + i\epsilon}(ig) = \frac{-ig^2}{s-m^2+i\epsilon}$$
2. t-channel: $t=(p1-p3)^2$ $$i\mathcal{M}_t = \frac{-ig^2}{t-m^2+i\epsilon}$$
3. u-channel: $u=(p_1-p_4)^2$ $$i\mathcal{M}_u = \frac{-ig^2}{u - m^2 + i\epsilon}$$ With $$\left|\mathcal{M}\right|^2 = \left|\mathcal{M}_s + \mathcal{M}_t + \mathcal{M}u\right|^2,$$ we get the cross section $$\frac{d\sigma}{d\Omega}(\phi\phi\to\phi\phi) = \frac{g^4}{64\pi^2E^2\text{CM}}\left[\frac{1}{s-m^2}+\frac{1}{t-m^2}+\frac{1}{u-m^2}\right]^2$$ with $s,u,t$ the Mandelstam variables

In the case of an interaction with derivatives of the fields, e.g. $$\mathcal{L}\text{int} = \lambda\phi(\partial\mu\phi)(\partial^\mu\phi)$$ We get a factor of $-ip_\mu$ if a field is entering a vertex (beeing destroyed by the vertex) and a factor of $ip_\mu$ if stems form the vertex (is created by the vertex). This can be seen when writing down the momentum space representation of the field operator and acting the derivative on in $\partial_\mu \phi(x)$.

Note:

• Total derivatives $\partial_\mu j^\mu$ in the Lagrangian do not contribute in perturabion theory